Finite Group Databasebeta

← #37542011

Subgroups — #37542011

Order 23168.

1102 conjugacy classes of subgroups; only the 197 normal subgroups are shown.

normal subgroups only

This group has 1102 conjugacy classes of subgroups — too many to draw. Only its 197 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C24×Dic181C211yesmaximal, characteristic, complemented
2C2×#19153797C212yesmaximal, complemented
3#26041159C214yesmaximal, complemented
4#26041151C214yesmaximal, complemented
5#26041140C212yesmaximal, complemented
6#26041135C211yesmaximal, characteristic, complemented
7C2×#120×C181C211yesmaximal, characteristic
8C23×Dic181—28no—
9C23×Dic181—24no—
10C23×Dic181C2214yes—
11C23×Dic181C2212yes—
12#19153797—28no—
13C2×#15668428C2214yescomplemented
14C2×C22⋊Dic181C2212yes—
15C2×C22⋊Dic181C2214yes—
16C2×C22⋊Dic181C2214yescomplemented
17#19153723C2218yescomplemented
18C25×C181C2211yescharacteristic
19C22×D4×C181C2212yes—
20C22×D4×C181C2211yescharacteristic
21#120×C181—24no—
22C2×C22⋊C4×C181C2211yescharacteristic
23C2×C22⋊C4×C181C2212yes—
24C22×Dic181—232no—
25C22×Dic181—216no—
26C22×Dic181—24no—
27C22×Dic181—24no—
28C22×Dic181—28no—
29C22×Dic181D418yescomplemented
30C22×Dic181C2314yes—
31C22×Dic181C2314yes—
32#15668428—28no—
33C22⋊Dic181—28no—
34C22⋊Dic181—24no—
35C22⋊Dic181—28no—
36C24×C181—24no—
37C24×C181—28no—
38C24×C181D414yes—
39C24×C181C2311yescharacteristic
40C24×C181C2311yescharacteristic
41C24×C181C2312yes—
42C2×D4×C181—22no—
43C2×D4×C181—24no—
44C2×D4×C181—24no—
45C2×D4×C181—28no—
46C22×C4×C181C2312yes—
47C22×C4×C181C2311yescharacteristic
48C22⋊C4×C181—22no—
49C22⋊C4×C181—24no—
50C2×Dic181—232no—
51C2×Dic181—216no—
52C2×Dic181—24no—
53C2×Dic181—24no—
54C2×Dic181—24no—
55C2×Dic181—28no—
56C2×Dic181C2×D418yes—
57C23×C181—232no—
58C23×C181—28no—
59C23×C181—216no—
60C23×C181—24no—
61C23×C181—24no—
62C23×C181—24no—
63C23×C181—21no—
64C23×C181—22no—
65C23×C181—24no—
66C23×C181C2×D414yes—
67C23×C181C2×D412yes—
68C23×C181C2×D414yes—
69C23×C181#4118yes—
70C23×C181C2411yescharacteristic
71D4×C181—44no—
72D4×C181—48no—
73C2×C4×C181—21no—
74C2×C4×C181—22no—
75C2×C4×C181—22no—
76C2×C4×C181—24no—
77C22×C181—232no—
78C22×C181—24no—
79C22×C181—24no—
80C22×C181—24no—
81C22×C181—28no—
82C22×C181—216no—
83C22×C181—41no—
84C22×C181—42no—
85C22×C181—44no—
86C22×C181#12014yes—
87C22×C181#12318yescomplemented
88C22×C181C22×D412yes—
89C22×C181C22×D411yescharacteristic
90C22×C181C2×#4114yes—
91C4×C181—41no—
92C4×C181—42no—
93Dic181—28no—
94Dic181—44no—
95C2×C181—24no—
96C2×C181—28no—
97C2×C181—81no—
98C2×C181C2×#12011yescharacteristic
99C2×C181C2×#12312yes—
100C2×C181#52514yes—
101C181#284911yescharacteristic, complemented, Sylow
102#2849—1811nomaximal, Sylow
103C24×C4—1811no—
104C2×#123—1812no—
105C2×#120D18111yescharacteristic
106#386—1814no—
107#385—1814no—
108#381—1812no—
109#379—1811no—
110C25C2×D18111yescharacteristic
111C22×D4C2×D18111yescharacteristic
112C22×D4C2×D18112yes—
113C23×C4—3624no—
114C23×C4—3628no—
115C23×C4—1814no—
116C23×C4—1812no—
117#123—3628no—
118#120—24no—
119C2×C4⋊C4—1814no—
120C2×C22⋊C4—1812no—
121C2×C22⋊C4—1814no—
122C2×C22⋊C4—1814no—
123C2×C22⋊C4C2×D18111yescharacteristic
124C2×C22⋊C4C2×D18112yes—
125#95—1818no—
126C24—24no—
127C24—28no—
128C24C22⋊D18114yes—
129C24C22×D18111yescharacteristic
130C24C22×D18111yescharacteristic
131C24C22×D18112yes—
132C2×D4—24no—
133C2×D4—22no—
134C2×D4—24no—
135C2×D4—28no—
136C22×C4—36232no—
137C22×C4—3624no—
138C22×C4—3628no—
139C22×C4—3624no—
140C22×C4—36216no—
141C22×C4—1814no—
142C22×C4—1818no—
143C22×C4—1814no—
144C22×C4C22×D18112yes—
145C22×C4C22×D18111yescharacteristic
146C4⋊C4—3628no—
147C22⋊C4—22no—
148C22⋊C4—24no—
149C22⋊C4—3624no—
150C22⋊C4—3628no—
151C22⋊C4—3628no—
152C23—232no—
153C23—28no—
154C23—216no—
155C23—24no—
156C23—24no—
157C23—24no—
158C23—22no—
159C23—21no—
160C23—24no—
161C23D4×D18114yes—
162C23C2×C22⋊D18112yes—
163C23C2×C22⋊D18114yes—
164C23#1566844818yes—
165C23C23×D18111yescharacteristic
166D4—44no—
167D4—48no—
168C2×C4—22no—
169C2×C4—21no—
170C2×C4—22no—
171C2×C4—24no—
172C2×C4—36232no—
173C2×C4—36216no—
174C2×C4—3624no—
175C2×C4—3624no—
176C2×C4—3624no—
177C2×C4—3628no—
178C2×C4—1818no—
179C22—232no—
180C22—24no—
181C22—24no—
182C22—24no—
183C22—28no—
184C22—216no—
185C22—41no—
186C22—42no—
187C22—44no—
188C22#1915378414yes—
189C22#1915379718yescomplemented
190C22C2×D4×D18112yes—
191C22C22×C22⋊D18111yescharacteristic
192C22C2×#1566844814yes—
193C4—41no—
194C4—42no—
195C4—3628no—
196C4—7244no—
197C2—24no—
198C2—28no—
199C2—81no—
200C2C2×#1915378411yescharacteristic
201C2C2×#1915379712yes—
202C2#2604178514yes—
subgroupG/H
center, FrattiniC23C23×D181
commutator, socle, cosocleC23×C181C24
Fitting, prime radicalC2×#120×C181C2
hypercenterC2×#120D181
hypocenterC181#2849
LCSG ▷ C23×C181 ▷ C181
UCSC1 ◁ C23 ◁ C2×#120

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C24×Dic181C2semidirect
2. C2×#19153797C2semidirect
3. #26041159C2semidirect
4. #26041151C2semidirect
5. #26041140C2semidirect
6. #26041135C2semidirect
13. C2×#15668428C22semidirect
16. C2×C22⋊Dic181C22semidirect
17. #19153723C22semidirect
29. C22×Dic181D4semidirect
87. C22×C181#123semidirect
101. C181#2849semidirect
189. C22#19153797semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C24×Dic1812
C2×#191537972
#260411592
#260411512
#260411402
#260411352
C2×#120×C1812
#2849181